Question

1) In a recent poll, the Gallup organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. Answer the questions below, showing work. Bare answers are not acceptable. (Showing work means writing the calculator command you used with correct input values in the correct order.)

f) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values in the United States is poor? Why or why not?

g) Now based on a random sample of 500 adult Americans, compute the mean and standard deviation of the random variable X, the number of adults who believe the overall state of moral values in the United States is poor. Interpret the mean.

h) Would it be unusual to identify 240 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans? Why? Now using the Normal Approximation to the Binomial, approximate the probability that:

i) Exactly 250 of those surveyed feel the overall state of moral values in the United States is poor.

j) Between 220 and 250, inclusive, of those surveyed feel the overall state of moral values in the United States is poor.

k) At least 260 adult Americans who believe the overall state of moral values in the United States is poor, based on a random sample of 500 adult Americans. Would this be unusual? Why?

Answer 1

1) n = 25

    p = 0.45

f) P(X > 20) = P(X = 20) + P(X = 21) + P(X = 22) + P(X = 23) + P(X = 24) + P(X = 25)

                   = 25C20 * (0.45)^20 * (0.55)^5 + 25C21 * (0.45)^21 * (0.55)^4 + 25C22 * (0.45)^22 * (0.55)^3 + 25C23 * (0.45)^23 * (0.55)^2 + 25C24 * (0.45)^24 * (0.55)^1 + 25C25 * (0.45)^25 * (0.55)^0 = 0.0004

Since the probability is less than 0.05, so itis unusual.
g) mean = n * p = 500 * 0.45 = 225

   Standard deviation = sqrt(np(1 - p))

                                  = sqrt(500 * 0.45 * 0.55)

                                  = 11.124

So the average no of adults who believe the overall state of moral values in the United states is poor is 225.

h)i) P(X = 250)

= P(249.5 < X < 250.5)

= P((249.5 - )/< (X - )/< (250.5 - )/)

= P((249.5 - 225)/11.124 < Z < (250.5 - 225)/11.124)

= P(2.2 < Z < 2.29)

= P(Z < 2.29) - P(Z < 2.2)

= 0.9890 - 0.9861

= 0.0029

Since the probability is less than 0.05, so it is unusual.
ii) P(220 < X < 250)

= P(219.5 < X < 250.5)

= P((219.5 - )/< (X - )/< (250.5 - )/)

= P((219.5 - 225)/11.124 < Z < (250.5 - 225)/11.124)

= P(-0.49 < Z < 2.29)

= P(Z < 2.29) - P(Z < -0.49)

= 0.9890 - 0.3121

= 0.6769

k) P(X > 260)

= P(X > 259.5)

= P((X - )/> (259.5 - )/)

= P(Z > (259.5 - 225)/11.124)

= P(Z > 3.1)

= 1 - P(Z < 3.1)

= 1 - 0.9990

= 0.0010

Since the probability is less than 0.05, so it is unusual.